[Math] Fractional derivative of a constant (Riemann-Liouville Derivative)

Question

In a book I read about Riemann-Liouville fractional derivative, it says,
$$_0D_t^\alpha 1=\frac{t^{-\alpha}}{\Gamma(1-\alpha)},\alpha\geq0,t\geq0$$ which identically vanishes for $\alpha\in\mathbb{N}$, due to the poles of Gamma function.(????)

I have two questions, the first is, why will the equation be applicable for $\alpha\geq0$? Isnt it that gamma function is only defined for positive arguments? Because i was thinking why the restriction is not $0<\alpha<1$.

My second question is what does that phrase "which identically vanishes for $\alpha\in\mathbb{N}$, due to the poles of Gamma function" mean.

Thank you.

Answer 1

1. The $\Gamma$ function is defined in the whole complex plane except for its simple poles at non positive integers.
   Here is a representation of the absolute value of $\Gamma$ :

2. The reciprocal of $\Gamma$ is an entire function with zeros for non positive integers.